1/17

EECE 301 Signals & SystemsProf. Mark Fowler

Note Set #19• D-T Signals: DTFT Details

2/17

DTFT Details

dtetxX tj )()(

In rad/sec radians

Compare to CTFT:

nj

nenxX

][)(

DT frequency in rad/sample

rad = (rad/sample) “sample”

Define the DTFT:

Very similar structure… so we should expect similar properties!!!

What we saw: That the conceptual CTFT inside the DAC can also be computed from the samples… we called that thing the DTFT.

3/17

Example of Analytically Computing the DTFT

n

][nx

-3 -2 -1 1 2 3 4 5 6

With your brain, not a computer

qn

qnan

nx n

,00,

0,0][

q = 3

oscillates ][,0

explodes"" ][,1

decays ][,1

nxaIf

nxaIf

nxaIf

By definition:

n

njenxX ][)(

q

n

njq

n

njn aeea00

)(

Given this signal model, find the DTFT.

j

qj

aeaeX

11)(

1

rrrr

qqq

qn

n

1

1212

1

General Form for Geometric Sum:

4/17

Characteristics of DTFT1.Periodicity of X()

X() is a periodic function of with period of 2

)()2( XX Recall pictures in notes of “DTFT Intro”:

Note: the CTFT does nothave this property

2. X() is complex valued (in general)

n

njenxX ][)( complex

Usually think of X() in polar form:)()()( XjeXX

magnitudephase Same

as CTFT

|X()| is periodic with period 2

X() is periodic with period 2

5/17

3. Symmetry

If x[n] is real-valued, then:

)()( XX

)()( XX

(even symmetry)

(odd symmetry)

Same as CTFT

Inverse DTFT

Q: Given X() can we find the corresponding x[n]?

A: Yes!!

deXnx jn)(

21][

We can integrate instead over any interval of length 2

…because the DTFT is periodic with period 2

6/17

Generalized DTFT

Periodic D-T signals have DTFT’s that contain delta functions

elsewhereperiodic

Xnnx,

),(2)(,1][

With a period of 2

X()2 2222

2 2 33

Main part

k

kX )2(2)(

Another way of writing this is:

Example:

7/17

How do we derive the result? Work backwards!

deXnx jn)(

21][

de jn)(2

21

0 jne

1

Sifting property

8/17

Transform Pairs: Like for the CTFT, there is a table of common pairs (See Web)

Be familiar with them Compare and contrast them with the tableOf common CTFT’s

9/17

DTFT of a Rectangular Pulse

otherwise

qqnnpq

,0

,,1,0,1,,,1][ as pulse T- D:Define

Subscript tells how far “left and right”

So, by DTFT definition:

q

qn

jnq eP )(

Use “Geometric Sum” Result…

2/sin

)2/1(sin1

)()1(

qeeeP j

qjjq

q

See book for details

10/17

Properties of the DTFT (See table provided)Like for the CTFT, there are many properties for the DTFT. Most are identical to those for the CTFT!!

But Note: “Summation Property” replaces Integration

There is no “Differentiation Property”

Compare and contrast these with the table of CTFT properties

Most important ones:

-Time shift

-Multiplication by sinusoid… Three “flavors”

-Convolution in the time domain

-Parseval’s Theorem

11/17

This one has no equivalent on CTFT Properties Table…

See next exampleIt provides a way to use a CTFT table to find DTFT pairs… here is an example

Comparing Properties of DTFT & CTFT

12/17

Example: Finding a DTFT pair from a CTFT pair

2--2

X()

Say we are given this DTFT and want to invert it…

The four steps for using “Relationship to Inverse CTFT” property are:

1. Truncate the DTFT X() to the - to range and set it to zero elsewhere2. Then treat the resulting function as a function of … call this ()

B-B

2--2

() = X()p2()

B-B

() = X()p2()

13/17

4. Get the x[n] by replacing t by n in (t)

From CTFT table:

tBBt

sinc)(

nBBtnx

nt sinc)(][

3. Find the inverse CTFT of () from a CTFT table, call it (t)

14/17

Example of DTFT of sinusoid

?)()cos(][ 0 Xnnx

Note that: )cos(1][ 0nnx

elsewhereperiodic2π

),(2)(

Y

So… use the “mult. by sinusoid” property

1][ ny

k

kY )2(2)( From DTFT

Table

Another way of writing this:

Y()

k = 0term

k = 1term

k = 2term

k = -2term

k = -1term

2 3 4--2-3-4

15/17

)()(21)( 00 YYX “mult. by

sinusoid” property says we shift up & down by 0

elsewhereperiodicX

2

,)()()(

00

Recall: )cos(1][ 0nnx so we can use the “mult. by sinusoid” result

Using the second form for Y() gives:

Or…using the first form for Y() gives:

k

kkY )2()2()( 00

16/17

To see this graphically:

elsewhereperiodic2π

),(2)(

Y

elsewhereperiodicX

2

,)()()(

00

0-0

Red comes from Up-shifted Y()Blue comes from Down-shifted Y()

X()

2 3 4--2-3-4

Y()

2 3 4--2-3-4

0

17/17

Comment on Some DTFT Forms on the Table

The last four entries on the DTFT Pairs Table are:

• Note that each of them has a summation… where the summation just adds in terms that are shifted by 2

• Note that because of this shift, only the k = 0 term lies between – and • Thus… we could more simply state these by writing only the k = 0 term

and stating that the result is 2-periodic elsewhere… like this:

nBB sinc 2 ( ),

2 -periodic elsewhereBp